Learn French: Tu es sûr qu'elle a eu une promotion ? - Are you sure that she had a promotion?

In French, when the conjunction que (that) is followed by a word starting with a vowel, like elle, the e is dropped and replaced with an apostrophe. This makes it flow better when spoken.

In English, we often drop the word 'that' and just say 'Are you sure she had a promotion?'. Can I drop qu' in French?

No, you can never drop que (or qu') in French. Even though 'that' is optional in English, you must always use que to connect the two clauses together in French.

How does a eu translate to 'had'?

This is the past tense (passé composé) of the verb avoir (to have). It uses avoir as its own helper verb (a) followed by its irregular past participle, which is eu (had).

Would the word sûr change if I were asking a woman?

Yes! The spelling sûr is masculine. If you are asking a woman 'Are you sure?', you need to add an e to make it feminine: Tu es sûre qu'elle a eu une promotion ? The pronunciation stays exactly the same.